Between your third and fourth lines, you use a√b√=ab−−√ . This is only (guaranteed to be) true when a≥0 and b>0 .

a√b√=ab−−√ has domain a≥0 and b>0 . Outside that domain, applying the identity is inappropriate, whether or not it "works."

In general (and this is the crux of most "fake" proofs involving square roots of negative numbers),x√ where x is a negative real number (x<0 ) must first be rewritten as i|x|−−√
before any other algebraic manipulations can be applied (because the
identities relating to manipulation of square roots [perhaps
exponentiation with non-integer exponents in general] require
nonnegative numbers).

This similar question, focused on−1=i2=(−1−−−√)2=−1−−−√−1−−−√=!−1⋅−1−−−−−−√=1√=1 , is using the similar identity a√b√=ab−−√ , which has domain a≥0 and b≥0 , so applying it when a=b=−1 is invalid.

*: As pointed out in the comments, what I meant was that the identity***edit**In general (and this is the crux of most "fake" proofs involving square roots of negative numbers),

This similar question, focused on

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